In this article, we will explore the concept of perpendicular lines thoroughly, starting from fundamental principles and progressing to detailed methods and examples. By the end, you'll have a comprehensive understanding of how to determine a line that is perpendicular to another, regardless of the form in which the original line is presented.
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Understanding Perpendicular Lines
What Does It Mean for Lines to Be Perpendicular?
Perpendicular lines are lines that intersect at a right angle (90 degrees). In a coordinate plane, if two lines are perpendicular, the angle between them is exactly 90 degrees. This geometric relationship has algebraic implications, especially when considering their slopes.
Key property:
- If two lines are perpendicular, the product of their slopes is -1 (assuming neither line is vertical).
Mathematically:
\[ m_1 \times m_2 = -1 \]
where \( m_1 \) and \( m_2 \) are the slopes of the two lines.
Note: This property applies when the lines are neither vertical nor horizontal. Special cases involving vertical or horizontal lines require additional considerations.
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Finding a Perpendicular Line: The Main Approaches
The process of finding a line perpendicular to a given line generally involves the following steps:
1. Identify the slope of the original line.
2. Determine the slope of the perpendicular line using the negative reciprocal.
3. Use a point (if given) or other known information to write the equation of the perpendicular line.
Let's explore each approach in detail.
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Step 1: Find the Slope of the Original Line
Depending on how the original line is presented, the method to find its slope varies.
Given the Equation in Slope-Intercept Form (y = mx + b)
- The slope is directly given as \( m \).
Given the Equation in Standard Form (Ax + By = C)
- Rearrange to slope-intercept form to identify the slope:
\[ y = -\frac{A}{B}x + \frac{C}{B} \]
- Slope: \( m = -\frac{A}{B} \).
Given Two Points \((x_1, y_1)\) and \((x_2, y_2)\)
- Calculate the slope as:
\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]
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Step 2: Determine the Slope of the Perpendicular Line
Once the original slope \( m \) is known, the slope \( m_{\perp} \) of the perpendicular line is:
- Negative reciprocal of \( m \):
\[ m_{\perp} = -\frac{1}{m} \]
Special Cases:
- If the original line is vertical (\( x = a \)), its slope is undefined, and the perpendicular line will be horizontal (\( y = b \)).
- If the original line is horizontal (\( y = c \)), its slope is zero, and the perpendicular line will be vertical (\( x = a \)).
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Step 3: Write the Equation of the Perpendicular Line
After finding the slope of the perpendicular line, you need a point through which this line passes to determine its equation.
Common scenarios:
- Given a point \((x_0, y_0)\):
- Use point-slope form:
\[ y - y_0 = m_{\perp}(x - x_0) \]
- Convert to slope-intercept or standard form as needed.
- Given the original line and a point on it:
- Use the point where the perpendicular line intersects the original line or any other specified point.
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Practical Examples
Example 1: Find the line perpendicular to \( y = 2x + 3 \) passing through \((4, 1)\)
Step 1: Identify the slope of the original line:
\[ m = 2 \]
Step 2: Find the slope of the perpendicular line:
\[ m_{\perp} = -\frac{1}{2} \]
Step 3: Use the point \((4, 1)\) and point-slope form:
\[ y - 1 = -\frac{1}{2}(x - 4) \]
Step 4: Simplify to slope-intercept form:
\[
\begin{aligned}
y - 1 &= -\frac{1}{2}x + 2 \\
y &= -\frac{1}{2}x + 3
\end{aligned}
\]
Answer: The perpendicular line is \( y = -\frac{1}{2}x + 3 \).
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Example 2: Find a perpendicular line to the line \( 3x - 4y = 7 \) passing through \((1, 2)\)
Step 1: Convert to slope-intercept form:
\[
\begin{aligned}
3x - 4y &= 7 \\
-4y &= -3x + 7 \\
y &= \frac{3}{4}x - \frac{7}{4}
\end{aligned}
\]
- Slope of original line:
\[ m = \frac{3}{4} \]
Step 2: Slope of perpendicular line:
\[
m_{\perp} = -\frac{1}{m} = -\frac{1}{\frac{3}{4}} = -\frac{4}{3}
\]
Step 3: Equation using point \((1, 2)\):
\[
y - 2 = -\frac{4}{3}(x - 1)
\]
Step 4: Simplify:
\[
\begin{aligned}
y - 2 &= -\frac{4}{3}x + \frac{4}{3} \\
y &= -\frac{4}{3}x + \frac{4}{3} + 2 \\
y &= -\frac{4}{3}x + \frac{4}{3} + \frac{6}{3} \\
y &= -\frac{4}{3}x + \frac{10}{3}
\end{aligned}
\]
Answer: The perpendicular line is \( y = -\frac{4}{3}x + \frac{10}{3} \).
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Special Cases in Finding Perpendicular Lines
While the general method involves using slopes and point-slope form, certain special cases require particular attention.
Vertical and Horizontal Lines
- Vertical line: \( x = a \). Any perpendicular line must be horizontal: \( y = b \).
- Horizontal line: \( y = c \). The perpendicular line must be vertical: \( x = d \).
Example:
- Given \( x = 5 \), the perpendicular line passing through any point not on \( x=5 \) is \( y = y_0 \).
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Using Coordinate Geometry and Vector Methods
Beyond slope calculations, other methods can be employed to find perpendicular lines, especially in vector form or when dealing with more complex geometric problems.
Dot Product Method
- The dot product of two perpendicular vectors is zero.
- If the direction vector of the original line is \( \vec{v} = (a, b) \), then a perpendicular vector \( \vec{v}_{\perp} \) satisfies:
\[ \vec{v} \cdot \vec{v}_{\perp} = 0 \]
- To find a perpendicular line passing through a point, select \( \vec{v}_{\perp} \) and write the parametric equations:
\[
\begin{cases}
x = x_0 + t \cdot a' \\
y = y_0 + t \cdot b'
\end{cases}
\]
where \( (a', b') \) is the perpendicular direction vector.
Example:
- Original line direction vector: \( (1, 2) \).
- Perpendicular vector: \( (-2, 1) \) (since \( 1 \times -2 + 2 \times 1 = 0 \)).
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Applications of Finding Perpendicular Lines
Understanding how to find perpendicular lines is fundamental in various fields:
- Geometry and Trigonometry: Solving problems involving angles, distances, and constructions.
- Analytical Geometry: Determining relationships between lines in the coordinate plane.
- Engineering and Design: Creating perpendicular components or structures.
Frequently Asked Questions
How do you find the equation of a line perpendicular to a given line?
First, find the slope of the given line. Then, take the negative reciprocal of that slope to get the perpendicular slope. Use this slope and a point on the new line to write its equation, often using point-slope form.
What is the slope of a line perpendicular to a line with slope m?
The slope of a line perpendicular to a line with slope m is the negative reciprocal, which is -1/m, provided m is not zero.
How do you determine if two lines are perpendicular?
Two lines are perpendicular if their slopes are negative reciprocals of each other. That is, the product of their slopes equals -1.
Can you find a perpendicular line passing through a specific point?
Yes. Find the slope of the original line, determine its negative reciprocal, then use the point-slope form with the given point and this slope to write the equation of the perpendicular line.
What is the difference between perpendicular and parallel lines in terms of their slopes?
Parallel lines have equal slopes, while perpendicular lines have slopes that are negative reciprocals of each other.
